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This is a comment to Sunil's answer on Euclid's proof of the infinitude of primes but I don't have enough point to leave a comment. There is a generalization of Euclid's proof which should be well known but I seldom see it mentioned. If $a_1,...,a_r$ are pairwise relatively prime integers, then for any subset |
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This is a comment to Sunil's answer on Euclid's proof of the infinitude of primes but I don't have enough point to leave a comment. There is a generalization of Euclid's proof which should be well known but I seldom see it mentioned. If $a_1,...,a_r$ are pairwise relatively prime integers, then for any subset $I \subset {1,...,r}$, $a(r+1)=\prod_{i \in I}a_i +\prod_{i \not in I}ai$not\in I}a_i$, is relatively prime to all the $a1,...,an$. |
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This is a comment to Sundil's Sunil's answer on Euclid's proof of the infinitude of primes but I don't have enough points point to leave a comment. There is a generalization of Euclid's proof which should be well known but I seldom see it mentioned. If $p1,...,pr$ are distinct primes, then for any subset $I \subset {1,...,r}$, and $P=\prod_{i \in I}pi + \prod_{i \not in I}pi$, (where empty product is 1) then $P$ is relatively prime to all the pi and hence must contain a prime divisor different from all the $pi$. Sunil's addition of a multiple is new to me and one can include this as $P=m\prod_{i \in I}pi + n\prod_{i \not in I}pi$, where m is relatively prime to pi for i \not in I and n is relatively prime to pi for i \in I. The same argument also implies by looking at prime divisor that if $a_1,...,a_r$ are pairwise relatively prime integers, then for any subset $I \subset {1,...,r}$, $a(r+1)=\prod_{i \in I}a_i +\prod_{i \not in I}ai$, is relatively prime to all the $a1,...,an$. |
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